Perfect Squares in Fibonacci-Product Bands: An Exact Length Spectrum and Minimum-Root Sequence

Abstract

Let F0=0,F1=1. For even m≥6 and L≥4, define I1(m,L)=[FmFm+L,FmFm+L+FL−2−1] and I2(m,L)=[FmFm+L+FL−1,FmFm+L+FL+1−1]. These intervals arise from Golden Exclusion choice words with a maximal two-copy stable period-4 tail C(⌊φf2⌋)=p(1100)r, r≥2. We prove that a square f2, with f not a Fibonacci number, occurs in the primitive bands exactly for L∈{5,13}∪{15,16,17,…}. The exclusions L=7,9,11 are computer-assisted: exact modular certificates eliminate 112 of 188 primitive corrections, leaving 76 genus-one quartics. Magma V2.29-9 certifies exact rank and the full Mordell-Weil group for all 76 residual curves, and no target solution remains. The least roots at the admissible lengths begin 70,183,296,378,479,611,775,986,… and, under the natural indexing, satisfy a(n)=⌈8Fn+18⌉(n≥7). The record includes the manuscript, source files, exact Python verifiers, Magma reproducibility files, SHA256 checksums, and a 10,000-term sequence file. The quartic and Fibonacci-numeration methods have substantial prior art; the claims here concern the exact classification proved in this work.